On perturbations of linear controllable infinite-dimensional time-varying discrete-time systems

It is shown that for a broad class of linear (possibly time-varying and infinite-dimensional) discrete-time systems x_k+1 = A_kx_k + B_ku_k the property of being uniformly equicontrollable is preserved under small perturbations of system parameters. The problem of controllability of asymptotically time-invariant systems is also studied.     

On ℓ∞ and L∞ gains for positive systems with bounded time-varying delays

This paper is devoted to the analysis of the ?_∞ and L_∞ gains for positive linear systems with interval time-varying delays. Through exploiting the monotonicity of the state trajectory, we first prove that for positive systems with constant delays, the ?_∞ and L_∞ gains are fully governed by the system matrices but independent of the delay size. Moreover, for positive systems with bounded time-varying delays, by comparing with the nominal systems with constant delays, it turns out that the ?_∞ and L_∞ gains are exactly the same as that of the constant delay systems. The results in this paper reveal that the ?_∞ and L_∞ gains of positive linear systems are not sensitive to the magnitude of time delays and hence the computation of ?_∞ and L_∞ gains of positive systems with bounded time-varying delays can be reduced to computing the ?_∞ and L_∞ gains of the corresponding delay-free systems. Both continuous-time and discrete-time cases are considered in this paper.     

Aspects of neural modeling of multidimensional chaotic attractors

This article discusses various aspects of neural modeling of multidimensional chaotic attractors. The Lorenz and Rosler attractors are considered as representative cases and are thoroughly examined. These two dynamical systems are expressed, within acceptable accuracy limits, by the corresponding systems of difference equations. Initially, the complete neural models of the attractors are examined. In this case, the neural networks are supplied with the values X_n , Y_n , Z_n of the systems to predict all the next components (X_{n +1}, Y_{n +1}, and Z_{n +1}) of the attractors. In the second case, named ‘component simulation’, the neural models are trained to predict only one of the values X_n , Y_n , Z_n , when they are fed with the complete input vector as in the first case. In the third case, the proposed neural networks are trained to predict only one component (X_{n +1}, Y_{n +1}, or Z_{n +1}) of the attractors, given a number of past values of the same component. Finally, the ability of the networks to predict the Y and Z components of an X time series of the dynamical systems is examined. Since the response of some networks is not satisfactory, the distribution of absolute error is considered in order to form a realistic picture of the networks’ performance.     

A parametrization for dissipative behaviors

The behavioral equivalent of single input single output (SISO) systems are behaviors with two manifest variables. Passive SISO systems can, therefore, be viewed as J-dissipative behaviors with two manifest variables. Here the special matrix J defines a QDF that captures the passivity property of SISO systems. In this paper, we investigate more general QDFs Q_Φs induced by some operator Φ. These QDFs define some relation between the input, the output and their derivatives of a SISO system. We characterize all behaviors that are dissipative with respect to the prescribed QDF Q_Φ. In fact, we parametrize all the behaviors dissipative with respect to Q_Φ in terms of those dissipative with respect to the special QDF Q_J induced by the matrix J. Similar results can also be given for lossless systems.     

A Piecewise Envelope Approach to H ∞ Control of Nonlinear Systems

This paper provides a computable approach for the H _∞ control synthesis of nonlinear systems using the piecewise envelope method. Firstly, the computation method of piecewise envelope is presented such that the required piecewise linear differential inclusions (PWLDIs) can be obtained by solving a linear optimization problem. Then, we propose the H _∞ control design method for PWLDIs using piecewise Lyapunov function theory, all the synthesis conditions are formulated as the feasibility of linear matrix inequalities, hence computationally tractable. Because of the inclusion relationship, the H _∞ controllers designed for PWLDIs are also effective for original nonlinear systems. The validity of the proposed approach is tested by application to control a wheeled mobile robot.     

Non‐fragile H ∞ control for switched stochastic delay systems with application to water quality process

In this paper, the problem of non‐fragile observer‐based H _∞ control for discrete‐time switched delay systems is investigated. Both data missing and time delays are taken into account in the links from sensors to observers and from controllers to actuators. Because data missing satisfies the Bernoulli distribution, such problem is transformed into an H _∞ control problem for stochastic switched delay systems. Average dwell time approach is used to obtain sufficient conditions on the solvability of such problems. A numerical example and a real example for water quality control are provided to illustrate the effectiveness and potential applications of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Local uncontrollability for affine control systems with jumps†

This paper investigates affine control systems with jumps for which the ideal I_f(g₁, …, g_m) generated by the drift vector field f in the Lie algebra L(f, g₁, …, g_m) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.     

Discrete-time linear periodic systems: A note on the reachability and controllability interval length

Discrete-time linear systems with periodic coefficients of period T are considered in this paper. The reachability and controllability indices at time t for periodic systems, μ_rt and μ_ct, are defined. It is shown that the reachability [controllability] subspace at time t does coincide with the reachability [controllability] subspace over the interval (t − μ_rt,T, t) [(t, t + μ_ct.,T)] and properly includes the reachability [controllability] subspace over the interval (t −(μ_rt−1) T, t) [(t, t +(μ_cr−1)T)].     

Improved Controller Design for Uncertain Positive Systems and its Extension to Uncertain Positive Switched Systems

This paper is concerned with the controller design of uncertain positive systems. First, we decompose the feedback gain matrix K_m×n into m×n non‐negative components and m×n non‐positive components. For the non‐negative components, each component contains only one positive element and the other elements are zero. Similarly, each non‐positive component contains only one negative element and the other ones are zero. Then, a simple and effective controller design approach of uncertain positive systems is proposed by incorporating the decomposed feedback gain matrix into the resulting closed‐loop systems and further applied to uncertain positive switched systems. It is shown that the designed controller is less conservative compared with those in the literature. Finally, a numerical example is provided to verify the validity of the proposed design.     

Dichotomy property and Lagrange stability for uncertain pendulum-like feedback systems

In this note, property of dichotomy and Lagrange stability of the uncertain pendulum-like systems with additional, multiplicative and H_∞ uncertainty are considered, respectively. By using some results of H_∞ theory the property of dichotomy and Lagrange stability of uncertain pendulum-like systems with multiple equilibria are transformed into L_∞ or H_∞ norm condition of some transfer functions. The convergence problem of all bounded solutions for some uncertain pendulum-like systems can be converted into the internal stability of another system under unstructured perturbations.     

Robust H∞ Sliding Mode Control for a Class of Singular Stochastic Nonlinear Systems

This paper concerns the problem of robust H_∞ sliding mode control for a class of singular stochastic nonlinear systems. Integral sliding mode control is developed to deal with this problem. Based on the integral sliding surface of the design and linear matrix inequality, a sufficient condition which guarantees the sliding mode dynamics is asymptotically mean square admissible and has a prescribed H_∞ performance for a class of singular stochastic nonlinear systems is proposed. Furthermore, a sliding mode control law is synthesized such that the singular stochastic nonlinear system can be driven to the sliding surface in finite time. Finally, a numerical example is proposed to illustrate the effectiveness of the given theoretical results.     

Robust filtering for a class of discrete-time uncertain nonlinear systems: An H∞ approach

This paper deals with the H_∞ filtering problem for a class of discrete-time nonlinear systems with or without real time-varying parameter uncertainty and unknown initial state. For the case when there is no parametric uncertainty in the system, we are concerned with designing a nonlinear H_∞ filter such that the induced l₂ norm of the mapping from the noise signal to the estimation error is within a specified bound. It is shown that this problem can be solved via one Riccati equation. We also consider the design of nonlinear filters which guarantee a prescribed H_∞ performance in the presence of parametric uncertainties. In this situation, a solution is obtained in terms of two Riccati equations.     

Simultaneous robust normalization and delay‐dependent robust H∞ stabilization for singular time‐delay systems with uncertainties in the derivative matrices

This paper investigates the problem of simultaneous robust normalization and delay‐dependent H_∞ control for a class of singular time‐delay systems with uncertainties. Not only the state and input matrices but also the derivative matrices of the considered systems are assumed to have uncertainties. New sufficient conditions for the existence of a proportional plus derivative state feedback H_∞ controller are derived as LMIs such that the closed‐loop singular system is normal, stable, and guarantee a specific level of performance. Specially, a static state feedback H_∞ controller alone or a state‐derivative feedback H_∞ controller alone can unite to be dealt with by applying our proposed method. Two simulation examples are provided to demonstrate the effectiveness of the proposed approach in this paper. Copyright © 2014 John Wiley & Sons, Ltd.     

Solving parametric fuzzy systems of linear equations by a nonlinear programming method

Linear systems of equations, with uncertainty on the parameters, play a major role in various problems in economics and finance. In this paper parametric fuzzy linear systems of the general form A ₁ x + b ₁ = A ₂ x + b ₂, with A ₁, A ₂, b ₁ and b ₂ matrices with fuzzy elements, are solved by means of a nonlinear programming method. The relation between this methodology and the algorithm proposed in Muzzioli and Reynaerts [(2006) Fuzzy Sets and Systems, in press] is highlighted. The methodology is finally applied to an economic and a financial problem.     

L1-gain analysis and control synthesis of positive switched systems

This paper investigates the problems of L₁-gain analysis and control synthesis of positive switched systems. Linear supply rates and L₁-gain notations are introduced to analyse the performance of the underlying systems. Stability with a weighted L₁-gain for autonomous systems are solved by using multiple linear copositive Lyapunov functions incorporated with the average dwell time approach. Then, state-feedback and output-feedback controllers are designed to guarantee the stabilisation with a weighted L₁-gain for non-autonomous systems. All present conditions are solvable in terms of linear programming. Finally, a practical example is given to illustrate the validity of theoretical findings.     

Feedback stabilization of real analytic systems in the plane

We investigate the local feedback stabilization of single input control affine analytic systems in the plane. New necessary and sufficient conditions for local stabilization with feedback laws of the form u = v(x₁,x₂), x2), (v/x₁(0, 0))² + (v/x₁ (0, 0))² 0≠ 0, v(0, 0) = 0, are obtained by using Lyapunov's stability theorems on two-dimensional analytic systems. If the sufficient conditions are satisfied, we also provide explicit feedback laws.     

Stabilization and H∞ Control of Nonlinear Switched Hamiltonian Systems Subject to Actuator Saturation

This paper investigates the stabilization and H_∞ control problem of nonlinear switched Hamiltonian systems (NSHSs) subject to actuator saturation (AS) under arbitrary switching paths. First, based on the saturating actuator property, an appropriate state feedback is designed under a realistic assumption for the stabilization of NSHSs with AS. Then, an H_∞ controller is designed for NSHSs subject to AS with external disturbances in order to attenuate the disturbances. Futhermore, the results obtained for NSHSs are applied to study the stabilization and H_∞ control of switched nonlinear systems with AS via the Hamiltonian realization method. Finally, simulation examples show the efficiency of the methods and results proposed in this paper.     

H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems

This paper concerns the problem of H_∞ filtering for piecewise homogeneous Markovian jump nonlinear systems. Different from the existing studies in the literatures, the existence of variations in transition rates for Markovian jump nonlinear systems is considered. The purpose of the paper is to design mode-dependent and mode-independent filters, such that the dynamics of the filtering errors are stochastic integral input-to-state stable with H_∞ performance index. Using the linear matrix inequality method and the Lyapunov functional method, sufficient conditions for the solution to the H_∞ filtering problem are derived. Finally, three examples are proposed to illustrate the effectiveness of the given theoretical results.